Explicit thresholds for approximately sparse compressed sensing via ℓ1-optimization

It is well known that compressed sensing problems reduce to solving large under-determined systems of equations. If we choose the elements of the compressed measurement matrix according to some appropriate probability distribution and if the signal is sparse enough then the lscr₁-optimization can recover it with overwhelming probability. In fact, establish (in a statistical context) that if the number of measurements is proportional to the length of the signal then there is a sparsity of the unknown signal proportional to its length for which the success of the lscr₁-optimization is guaranteed. In this paper we consider a modification of this standard setup, namely the case of the so-called approximately sparse unknown signals. We determine sharp lower bounds on the values of allowable approximate sparsity for any given number (proportional to the length of the unknown signal) of measurements. We introduce a novel, very simple technique which provides very good values for proportionality constants.